20 is divided into two parts so that product of cube of one quantity and square of the other quantity is maximum. The parts are

Options

A. 10, 10
B. 16, 4
C. 8, 12
D. 12, 8

Explanation

To solve the problem of dividing 20 into two parts \( x \) and \( y \) such that the product \( P = x^3 y^2 \) is maximized, we can follow these steps:

### Step 1: Set up the equations
We know from the problem that:
\[ x + y = 20 \]
This can be rearranged to express \( y \) in terms of \( x \):
\[ y = 20 - x \]

### Step 2: Substitute \( y \) into the product equation
Now, we substitute \( y \) into the product \( P \):
\[ P = x^3 (20 - x)^2 \]

### Step 3: Expand the product
Next, we expand the expression for \( P \):
\[ P = x^3 (400 - 40x + x^2) \]
\[ P = 400x^3 - 40x^4 + x^5 \]

### Step 4: Differentiate to find the maximum
To find the maximum value of \( P \), we need to take the derivative of \( P \) with respect to \( x \) and set it to zero:
\[ \frac{dP}{dx} = 1200x^2 - 160x^3 + 5x^4 \]
Setting the derivative equal to zero:
\[ 1200x^2 - 160x^3 + 5x^4 = 0 \]

### Step 5: Factor the equation
We can factor out \( x^2 \):
\[ x^2 (1200 - 160x + 5x^2) = 0 \]
This gives us one solution \( x = 0 \) (which we discard as it does not make sense in this context). We need to solve the quadratic:
\[ 5x^2 - 160x + 1200 = 0 \]

### Step 6: Use the quadratic formula
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 5 \), \( b = -160 \), and \( c = 1200 \):
\[ x = \frac{160 \pm \sqrt{(-160)^2 - 4 \cdot 5 \cdot 1200}}{2 \cdot 5} \]
\[ x = \frac{160 \pm \sqrt{25600 - 24000}}{10} \]
\[ x = \frac{160 \pm \sqrt{1600}}{10} \]
\[ x = \frac{160 \pm 40}{10} \]

Calculating the two possible values:
1. \( x = \frac{200}{10} = 20 \)
2. \( x = \frac{120}{10} = 12 \)

### Step 7: Find corresponding \( y \) values
For \( x = 12 \):
\[ y = 20 - 12 = 8 \]

For \( x = 20 \):
\[ y = 20 - 20 = 0 \] (not valid)

### Conclusion
Thus, the two parts that maximize the product \( P = x^3 y^2 \) are:
\[ x = 12 \quad \text{and} \quad y = 8 \]

### Final Answer
The parts are \( 12 \) and \( 8 \).

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